Functional Observability of Networks

Functional observability analysis, minimum sensor placement, and minimum-order functional observer design for target state estimation in large-scale dynamical networks, including applications to power grids and epidemics.

License

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

The full text of the GNU General Public License can be found in the file "LICENSE.txt".

Usage

The following codes are direct implementations of the minimum sensor placement algorithm and the minimum-order functional observer design proposed in Ref. [1].

• find_msp : Finds the minimum set of sensor nodes required for the system functional observability with respect to a set of target nodes. This is a MATLAB implementation of Algorithm 1 in Ref. [1].

• find_F0 : Finds F0 with minimum-order such that Darouach's condition (4) in Ref. [1] is satisfied for a triple (A,C,F0). This is a MATLAB implementation of Algorithm 2 in Ref. [1].

• functobsv_design : Designs the functional observer's matrices (N,J,H,D,E) in Eq. (10) of Ref. [1]. The design method is guaranteed to provide a stable functional observer if the triple (A,C,F0) satisfies conditions (4-5) in Ref. [1].

The following examples illustrate numerical results of the algorithms described above in applications to large-scale complex networks, power grids, and epidemics. For more details, see the full description in each file.

• example_dynamicalnetwork : Example of minimum sensor placement and minimum-order functional observer design for a random complex dynamical networks. This code examplifies how to:

  1. apply Algorithm 1 to determine the minimum set of sensor nodes for functional observability of a dynamical network with respect to a given set of target nodes; and
  2. apply Algorithm 2 to design a minimum-order functional observer. Both algorithms are further described in Ref. [1] and in codes find_msp.m and find_F0.m.

• example_cyberdetection : Example of cyber-attack detection in power grids using functional observers and target state estimation.

• example_epidemicspreading : Example of sensor placement and nonlinear functional observer design for estimating the prevalence rate of infection in target populations following an epidemic outbreak.

The above examples are dependent on the following codes:

• odeRK : Implements the numerical integration algorithm Runge Kutta 4th order.
• functobsv_sys,functobsv_nonlinearsys : Represents the ODEs of a linear functional observer (Ref. [1], Eq. (10)) and a nonlinear functional observer (Ref. [1], Eq. (S-24)), respectively.
• spnull,sporth : Computes a sparse orthonormal basis for the null space and the range space of a matrix, respectively.

References

1. A. N. Montanari, C. Duan, L. A. Aguirre, A. E. Motter. Functional observability and target state estimation in large-scale networks. Proceedings of the National Academy of Sciences 119(1):e2113750119 (2022). https://doi.org/10.1073/pnas.2113750119